Article Contents
Article Contents

# Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit

• * Corresponding author: Yu-Chung Tsao

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Conceptual model for deteriorating products under order-linked trade credit

Figure 2.  Inventory level

Figure 3.  Graphical representation of Sub-case 1.1

Figure 4.  Graphical representation of Sub-case 1.2

Figure 5.  Graphical representation of Sub-case 2.1

Figure 6.  Graphical representation of Sub-case 2.2

Figure 7.  The graph of $PTP^{}_{1.1}(p, M^{}, T^{})$

Figure 8.  The graph of $PTP^{}_{1.1}(p, M^{}, T^{})$

Figure 9.  The graph of $PTP^{}_{1.1}(p, M^{}, T^{})$

Table 1.  Summary of literature review

Table 2.  Notations

 Notation Description $i$ Index of case based on demand, $i=\lbrack 1, 2\rbrack$ $j$ Index of case based on time, $j=\lbrack 1, 2\rbrack$ $A$ Replenishment cost per order, in dollar $c$ Procurement cost per unit, in dollar $h$ The inventory holding cost rate per unit per unit time, in dollar $I_{e}$ Interest earned per dollar per unit time $I_{Loss}$ Interest revenue loss due to offering trade credit per dollar per unit time $r$ Annual compound interest rate per dollar per unit time $D(p, M)$ Annual demand rate per unit time, as a function of both $p$and $M$ $F(M)$ The rate of default risk $\theta$ Deterioration rate, $0\le \theta \le 1$ $D_{d}$ The specific threshold in which permits the full trade credit $\alpha$ The fraction of the delay payments is permitted, $0\le \alpha \le 1$ $I(t)$ Inventory level at time $t$ $Q$ Seller's order quantity $OC$ The present value of ordering cost $HC$ The present value of holding cost $PC$ The present value of procurement cost $SR_{i}$ The present value of revenue in case i, $i=\lbrack 1, 2\rbrack$ $IE_{i, j}$ The present value of interest earned in case (i, j), $i, j=\lbrack 1, 2\rbrack$ $IL_{i, j}$ The present value of interest loss in case (i, j), $i, j=\lbrack 1, 2\rbrack$ $T$ Length of the replenishment cycle (decision variable) $M$ The credit period policies offered by the seller (decision variable) $p$ Selling price offered by the seller per unit (decision variable) $PTP(p, M, T)$ The present value of total annual profit, which is the function of $p$, $M$ and $T$

Table 3.  Interest earned and interest revenue loss cases

 Case (1):$D_{d} Table 4. A comparison of four cases:  Cases$ p^{} $(＄)$ M^{} $(years)$ T^{} $(years) Demand (items)$ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $(＄) Ordered-link trade credit 25.3312 0.081921 0.27852 375 5482.41 No ordered-link trade credit 26.2325 0.25352 344 5398.24 Full trade credit 25.505 0.054 0.27741 368 5478.92 Partial trade credit 25.7267 0.151 0.2979 372 5485.39 Table 5. Summary of sensitivity analysis  Parameter$ p^{}  M^{}  T^{}  PTP^{}_{i.j}(p^{}, M^{}, T^{}) $D$ _{d} $=350 25.3312 0.0819213 0.266878 5482.41 D$ _{d} $=450 25.5224 0.149572 0.297398 5493.47 c=6 23.7634 0.260868 0.360191 7123.23 c=8 24.6262 0.196586 0.320289 6280.3 c=10 25.5224 0.149572 0.297398 5493.47 c=12 26.4370 0.112410 0.28441 4761.02 c=14 27.3616 0.081181 0.277903 4081.91 h=0.6 25.5873 0.177748 0.360788 5518.11 h=0.8 25.5501 0.161851 0.324659 5505.19 h=1 25.5224 0.149572 0.297398 5493.47 h=1.2 25.5012 0.139723 0.275927 5482.67 h=1.4 25.4845 0.131596 0.258471 5472.62$ \theta $=0.03 25.5515 0.162354 0.325811 5505.41$ \theta $=0.04 25.5359 0.155573 0.310662 5499.3$ \theta $=0.05 25.5224 0.149572 0.297398 5493.47$ \theta $=0.06 25.5106 0.144213 0.285662 5487.86$ \theta $=0.07 25.5002 0.139388 0.275185 5482.47 b=15 39.6903 0.291871 0.409115 11962.1 b=20 30.7678 0.198115 0.322033 7882.19 b=25 25.5224 0.149572 0.297398 5493.47 b=30 22.0502 0.114364 0.291447 3947.53 b=35 19.5771 0.084373 0.29072 2882.45 l=0.06 27.3895 0.790391 0.771212 5638.77 l=0.08 26.0368 0.328971 0.397466 5537.01 l=0.1 25.5224 0.149572 0.297398 5493.47 l=0.12 25.2051 0.0372622 0.268559 5476.34 l=0.14 25.1003 6.15627$ \times $10$ ^{-27} $0.266878 5475.06 Table 6. Summary of sensitivity analysis (cont')  Parameter$ p  M  T  PTP^_{i.j}(p^, M^, T^)  \alpha $=0.48 26.3120 0.427639 0.398781 5544.5$ \alpha $=0.64 25.7899 0.243915 0.329357 5510.66$ \alpha $=0.8 25.5224 0.149572 0.297398 5493.47$ \alpha $=0.96 25.3619 0.092792 0.281188 5484.03$ \alpha $=1 25.3312 0.081921 0.27852 5482.41 r=0.024 25.9844 0.307269 0.445592 5546.65 r=0.032 25.7022 0.211592 0.349793 5516.32 r=0.04 25.5224 0.149572 0.297398 5493.47 r=0.048 25.3915 0.103826 0.264839 5475.44 r=0.056 25.2871 0.067011 0.242921 5460.93 I$ _{Loss} $=0.036 25.7848 0.242724 0.314041 5504.49 I$ _{Loss} $=0.048 25.6232 0.185330 0.304028 5497.72 I$ _{Loss} $=0.06 25.5224 0.149572 0.297398 5493.47 I$ _{Loss} $=0.072 25.4539 0.125260 0.292728 5490.55 I$ _{Loss} $=0.084 25.4043 0.107692 0.289277 5488.43 A=12 25.4177 0.120489 0.232408 5523.65 A=16 25.4741 0.136219 0.26707 5507.64 A=20 25.5224 0.149572 0.297398 5493.47 A=24 25.5651 0.161264 0.324661 5480.61 A=28 25.6034 0.171715 0.349611 5468.74 n=60 25.1003 1.57412$ \times $10$ ^{-9} $0.266878 5475.06 n=80 25.1003 6.67337$ \times $10$ ^{-9} $0.266878 5475.06 n=100 25.5224 0.149572 0.297398 5493.47 n=120 26.6491 0.452890 0.489132 5575.1 n=140 29.2678 1.039800 101428 5783.31 I$ _{e} $=0.024 25.4698 0.131901 0.258682 5472.69 I$ _{e} $=0.032 25.4934 0.139910 0.276078 5482.71 I$ _{e} $=0.04 25.5224 0.149572 0.297398 5493.47 I$ _{e} $=0.048 25.5589 0.161542 0.324322 5505.12 I$ _{e} \$=0.056 25.6065 0.176891 0.359702 5517.93

Figures(9)

Tables(6)